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erdős #860

← #859 · #861 → (packet.json; erdosproblems.com)

Let $h (n)$ be such that, for any $m \geq 1$ , in the interval $(m, m + h (n))$ there exist distinct integers $a_{i}$ for $1 \leq i \leq π (n)$ such that $p_{i} ∣ a_{i}$ , where $p_{i}$ denotes the $i$ th prime. Estimate $h (n)$ .

Worked, still open.

number theory · open · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (P(n)={p\le n:\ p\ \text{prime}}); then (|P(n)|=\pi(n)), and the condition says:

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

oeis

A048670 — Jacobsthal function A048669 applied to the product of the first n primes (A002110).2,4,6,10,14,22,26,34,40,46,58,66,74,90,100,106,118,132,152,174,190,200,216,234,258,264,282,300,312,330,354,378,388,414,4 A058989 — Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.1,3,5,9,13,21,25,33,39,45,57,65,73,89,99,105,117,131,151,173,189,199,215,233,257,263,281,299,311,329,353,377,387,413,431

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Y (x)

y

such that there exists a choice of congruence classes

a_{p}

p \leq x

such that every integer in

[1, y]

is congruent to at least one of the

a_{p} (mod p)

. Give good estimates for

Y (x)

. In particular, can one prove that

Y (x) = o (x^{2})

Y (x) ≪ x^{1 + o (1)}

?A048670 #854Let

n_{k}

k

th primorial, i.e. the product of the first

k

1 = a_{1} < a_{2} < \dots a_{ϕ (n_{k})} = n_{k} - 1

is the sequence of integers coprime to

n_{k}

, then estimate the smallest even integer not of the form

a_{i + 1} - a_{i}

≫ i max (a_{i + 1} - a_{i})

many even integers of the form

a_{j + 1} - a_{j}

status

open

finding.noted · reviewer:will-blair · 1 day

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